Solving variational inequalities with monotone operators on domains given by Linear Minimization Oracles

Abstract

The standard algorithms for solving large-scale convex–concave saddle point problems, or, more generally, variational inequalities with monotone operators, are proximal type algorithms which at every iteration need to compute a prox-mapping, that is, to minimize over problem’s domain X the sum of a linear form and the specific convex distance-generating function underlying the algorithms in question. (Relative) computational simplicity of prox-mappings, which is the standard requirement when implementing proximal algorithms, clearly implies the possibility to equip X with a relatively computationally cheap Linear Minimization Oracle (LMO) able to minimize over X linear forms. There are, however, important situations where a cheap LMO indeed is available, but where no proximal setup with easy-to-compute prox-mappings is known. This fact motivates our goal in this paper, which is to develop techniques for solving variational inequalities with monotone operators on domains given by LMO. The techniques we discuss can be viewed as a substantial extension of the proposed in Cox et al. (Math Program Ser B 148(1–2):143–180, 2014) method of nonsmooth convex minimization over an LMO-represented domain.

Proofs

1.1 Proof of Theorems 1 and 2

We start with proving Theorem 2. In the notation of the theorem, we have

$$\begin{aligned} \begin{array}{lcl} &{}&{}\forall x\in X: \Phi (x)=Ay(x)+a,\\ (a):&{}&{}y(x)\in Y,\\ (b): &{}&{}\langle y(x)-y,A^*x-G(y(x))\rangle \ge 0\,\forall y\in Y. \end{array} \end{aligned}$$

(52)

For $\bar{x}\in X$, let $\bar{y}=y(\bar{x})$, and let $\widehat{y}=\sum \nolimits _t\lambda _ty_t$, so that $\bar{y},\widehat{y}\in Y$ by (52.a). Since $G$ is monotone, for all $t\in \{1,\ldots ,N\}$ we have

$$\begin{aligned} \begin{array}{lll} &{}&{}\langle \bar{y}-y_t,G(\bar{y})-G(y_t)\rangle \ge 0\\ \Rightarrow &{}&{}\langle \bar{y},G(\bar{y})\rangle \ge \langle y_t,G(\bar{y})\rangle +\langle \bar{y},G(y_t)\rangle -\langle y_t,G(y_t)\rangle \,\,\forall t\\ \Rightarrow &{} &{}\langle \bar{y},G(\bar{y})\rangle \ge \sum \limits _t\lambda _t\left[ \langle y_t,G(\bar{y})\rangle +\langle \bar{y},G(y_t)\rangle -\langle y_t,G(y_t)\rangle \right] \\ &{}&{}[\hbox {since}\,\, \lambda _t\ge 0 \,\,\hbox {and} \,\, \sum \limits _t\lambda _t=1], \end{array} \end{aligned}$$

and we conclude that

$$\begin{aligned} \langle \bar{y},G(\bar{y})\rangle -\langle \widehat{y},G(\bar{y})\rangle \ge \sum _{t=1}^N\lambda _t\left[ \langle \bar{y},G(y_t)\rangle -\langle y_t,G(y_t)\rangle \right] . \end{aligned}$$

(53)

We now have

$$\begin{aligned}&\langle \Phi (\bar{x}),\bar{x}-\sum \limits _t\lambda _tx_t\rangle \\&\quad =\langle A\bar{y}+a,\bar{x}-\sum \limits _t\lambda _tx_t\rangle =\langle \bar{y},A^*\bar{x}-\sum \limits _t\lambda _tA^*x_t\rangle +\langle a,\bar{x}-\sum \limits _t\lambda _tx_t\rangle \\&\quad =\langle \bar{y},A^*\bar{x}-G(\bar{y})\rangle +\langle \bar{y},G(\bar{y})-\sum \limits _t\lambda _tA^*x_t\rangle +\langle a,\bar{x}-\sum \limits _t\lambda _tx_t\rangle \\&\quad \ge \langle \widehat{y},A^*\bar{x}-G(\bar{y})\rangle +\langle \bar{y},G(\bar{y})-\sum \limits _t\lambda _tA^*x_t\rangle +\langle a,\bar{x}-\sum \limits _t\lambda _tx_t\rangle \\&\qquad [\hbox {by (52.b) with}\,\, y=\widehat{y}\,\, \hbox {and due to} \,\,\bar{y}=y(\bar{x})]\\&\quad =\langle \widehat{y},A^*\bar{x}\rangle +\left[ \langle G(\bar{y}),\bar{y}\rangle -\langle G(\bar{y}),\widehat{y}\rangle \right] -\langle \bar{y},\sum \limits _t\lambda _tA^*x_t\rangle +\langle a,\bar{x}-\sum \limits _t\lambda _tx_t\rangle \\&\quad \ge \langle \widehat{y},A^*\bar{x}\rangle +\sum \limits _t\lambda _t\left[ \langle \bar{y},G(y_t)\rangle -\langle y_t,G(y_t)\rangle \right] -\langle \bar{y},\sum \limits _t\lambda _tA^*x_t\rangle \\&\qquad +\langle a,\bar{x}-\sum \limits _t\lambda _tx_t\rangle \,\,[\hbox {by (53)}]\\&\quad =\sum \limits _t\lambda _t\langle y_t,A^*\bar{x}\rangle +\sum \limits _t\lambda _t\left[ \langle \bar{y},G(y_t)\rangle -\langle y_t,G(y_t)\rangle -\langle \bar{y},A^*x_t\rangle +\langle a,\bar{x}-x_t\rangle \right] \\&\qquad [\hbox {since}\,\, \widehat{y}=\sum \limits _t\lambda _ty_t\,\, \hbox {and}\,\, \sum \limits _t\lambda _t=1]\\&\quad =\sum \limits _t\lambda _t\left[ \langle Ay_t,\bar{x}-x_t\rangle +\langle Ay_t,x_t\rangle +\langle \bar{y},G(y_t)\rangle -\langle y_t,G(y_t)\rangle \right. \\&\qquad \left. -\langle \bar{y},A^*x_t\rangle +\langle a,\bar{x}-x_t\rangle \right] \\&\quad =\sum \limits _t\lambda _t\left[ \langle y_t,A^*x_t\rangle +\langle \bar{y},G(y_t)\rangle -\langle y_t,G(y_t)\rangle -\langle \bar{y},A^*x_t\rangle \right] \\&\qquad + \sum \limits _t\lambda _t\langle Ay_t+a,\bar{x}-x_t\rangle \\&\quad ={\sum }_t\lambda _t\langle A^*x_t-G(y_t),y_t-\bar{y}\rangle +\sum \limits _t\lambda _t\langle Ay_t+a,\bar{x}-x_t\rangle \ge -\epsilon \\&\qquad +\sum \limits _t\lambda _t\langle Ay_t+a,\bar{x}-x_t\rangle \\&\qquad [\hbox {by} (15) \,\,\hbox {due to} \bar{y}=y(\bar{x})\in Y(X)]. \end{aligned}$$

The bottom line is that

$$\begin{aligned} \langle \Phi (\bar{x}),\widehat{x}-\bar{x}\rangle \le \epsilon +\sum _{t=1}^N\lambda _t\langle Ay_t+a,x_t-\bar{x}\rangle \,\forall \bar{x}\in X, \end{aligned}$$

as stated in (16). Theorem 2 is proved.

To prove Theorem 1, let $y_t\in Y$, $1\le t\le N$, and $\lambda _1,\ldots ,\lambda _N$ be from the premise of the theorem, and let $x_t$, $1\le t\le N$, be specified as $x_t=x(y_t)$, so that $x_t$ is the minimizer of the linear form $\langle Ay_t+a,x\rangle$ over $x\in X$. Due to the latter choice, we have $\sum _{t=1}^N\lambda _t\langle Ay_t+a,x_t-\bar{x}\rangle \le 0$ for all $\bar{x}\in X$, while $\epsilon$ as defined by (15) is nothing but ${\mathrm{Res}}(\{y_t,\lambda _t,-\Psi (x_t)\}_{t=1}^N|Y(X))$. Thus, (16) in the case in question implies that

$$\begin{aligned} \forall \bar{x}\in X: \langle \Phi (\bar{x}),\sum \limits _{t=1}^N\lambda _t x_t - \bar{x}\rangle \le {\mathrm{Res}}(\{y_t,\lambda _t,-\Psi (x_t)\}_{t=1}^N|Y(X)), \end{aligned}$$

and (13) follows. Relation (14) is an immediate corollary of (13) and Lemma 2 as applied to $X$ in the role of $Y$, $\Phi$ in the role of $H(\cdot )$, and $\{x_t,\lambda _t,\Phi (x_t)\}_{t=1}^N$ in the role of $\mathcal{C}^N$. $\square$

1.2 Proof of Proposition 1

Observe that the optimality conditions in the optimization problem specifying $v={{\mathrm{Prox}}}_y(\xi )$ imply that

$$\begin{aligned} \langle \xi -\omega '(y)+\omega '(v),z-v\rangle \ge 0,\,\,\forall z\in Y, \end{aligned}$$

or

$$\begin{aligned} \langle \xi ,v-z\rangle \le \langle \omega '(v)-\omega '(y),z-v\rangle =\langle V'_y(v),z-v\rangle ,\,\,\forall z\in Y, \end{aligned}$$

which, using a remarkable identity [5]

$$\begin{aligned} \langle V'_y(v),z-v\rangle =V_y(z)-V_v(z)-V_y(v), \end{aligned}$$

can be rewritten equivalently as

$$\begin{aligned} v={{\mathrm{Prox}}}_y(\xi )\Rightarrow \langle \xi ,v-z\rangle \le V_y(z)-V_v(z)-V_y(v)\,\,\forall z\in Y. \end{aligned}$$

(54)

Setting $y=y_t$, $\xi =\gamma _t H_t(y_t)$, which results in $v=y_{t+1}$, we get

$$\begin{aligned} \forall z\in Y: \gamma _t\langle H_t(y_t),y_{t+1}-z\rangle \le V_{y_t}(z)-V_{y_{t+1}}(z)-V_{y_t}(y_{t+1}), \end{aligned}$$

whence,

$$\begin{aligned} \forall z\in Y: \gamma _t\langle H_t(y_t),y_{t}-z\rangle\le & {} V_{y_t}(z)-V_{y_{t+1}}(z)+\underbrace{\left[ \gamma _t\langle H_t(y_t),y_t-y_{t+1}\rangle -V_{y_t}(y_{t+1})\right] }_{\le \gamma _t\Vert H_t(y_t)\Vert _*\Vert y_t-y_{t+1}\Vert -\frac{1}{2}\Vert y_t-y_{t+1}\Vert ^2} \\\le & {} V_{y_t}(z)-V_{y_{t+1}}(z)+\frac{1}{2}\gamma _t^2\Vert H_t(y_t)\Vert _*^2. \end{aligned}$$

Summing up these inequalities over $t=1,\ldots ,N$ and taking into account that for $z\in Y'$, we have $V_{y_1}(z)\le \frac{1}{2}\Omega ^2[Y']$ and that $V_{y_{N+1}}(z)\ge 0$, we get (19). $\square$

1.3 Proof of Proposition 2

Applying (54) to $y=y_t$, $\xi =\gamma _t H_t(z_t)$, which results in $v=y_{t+1}$, we get

$$\begin{aligned} \forall z\in Y: \gamma _t\langle H_t(z_t),y_{t+1}-z\rangle \le V_{y_t}(z)-V_{y_{t+1}}(z)-V_{y_t}(y_{t+1}), \end{aligned}$$

whence, by the definition (23) of $d_t$,

$$\begin{aligned} \begin{array}{l} \forall z\in Y: \gamma _t\langle H_t(z_t),z_t-z\rangle \le V_{y_t}(z)-V_{y_{t+1}}(z)+d_t.\\ \end{array} \end{aligned}$$

(55)

Summing up the resulting inequalities over $t=1,\ldots ,N$ and taking into account that $V_{y_1}(z)\le \frac{1}{2}\Omega ^2[Y']$ for all $z\in Y'$ and $V_{y_{N+1}}(z)\ge 0$, we get

$$\begin{aligned} \forall z\in Y': \sum _{t=1}^n\lambda ^N_t\langle H_t(z_t),z_t-z\rangle \le {\frac{1}{2}\Omega ^2[Y']+\sum _{t=1}^N d_t\over \sum _{t=1}^N\gamma _t}. \end{aligned}$$

The right hand side in the latter inequality is independent of $z\in Y'$. Taking supremum of the left hand side over $z\in Y'$, we arrive at (24).

Moreover, invoking (54) with $y=y_t$, $\xi =\gamma _t H_t(y_t)$ and specifying $z$ as $y_{t+1}$, we get

$$\begin{aligned} \gamma _t\langle H_t(y_t),z_t-y_{t+1}\rangle \le V_{y_t}(y_{t+1})-V_{z_t}(y_{t+1})-V_{y_t}(z_t), \end{aligned}$$

whence

$$\begin{aligned} \begin{aligned} d_t&=\gamma _t\langle H_t(z_t),z_t-y_{t+1}\rangle - V_{y_t}(y_{t+1})\le \gamma _t\langle H_t(y_t),z_t-y_{t+1}\rangle \\&\quad +\gamma _t\langle H_t(z_t)-H_t(y_t),z_t-y_{t+1}\rangle - V_{y_t}(y_{t+1})\\&\le -V_{z_t}(y_{t+1})-V_{y_t}(z_t)+\gamma _t\langle H_t(z_t)-H_t(y_t),z_t-y_{t+1}\rangle \\&\le \gamma _t\Vert H_t(z_t)-H_t(y_t)\Vert _*\Vert z_t-y_{t+1}\Vert -{\frac{1}{2}}\Vert z_t-y_{t+1}\Vert ^2-{\frac{1}{2}}\Vert y_t-z_t\Vert ^2\\&\le {\frac{1}{2}}\left[ \gamma _t^2\Vert H_t(z_t)-H_t(y_t)\Vert _*^2-\Vert y_t-z_t\Vert ^2\right] ,\\ \end{aligned} \end{aligned}$$

(56)

as required in (25). $\square$

1.4 Proof of Lemma 3

1 $^0$. We start with the following standard fact:

Lemma 4

Let $Y$ be a nonempty closed convex set in Euclidean space $F$, $\Vert \cdot \Vert$ be a norm on $F$, and $\omega (\cdot )$ be a continuously differentiable function on $Y$ which is strongly convex, modulus 1, w.r.t. $\Vert \cdot \Vert$. Given $b\in F$ and $y\in Y$, let us set

$$\begin{aligned} \begin{array}{rcl} g_y(\xi )&{}=&{}\max \limits _{z\in Y}\left[ \langle z,\omega '(y)-\xi \rangle -\omega (z)\right] :F\rightarrow {\mathbf {R}}, \\ {{\mathrm{Prox}}}_y(\xi )&{}=&{}\mathop {\hbox {argmax }}\limits _{z\in Y}\left[ \langle z,\omega '(y)-\xi \rangle -\omega (z)\right] . \\ \end{array} \end{aligned}$$

The function $g_y$ is convex with Lipschitz continuous gradient $\nabla g_y(\xi )=-{{\mathrm{Prox}}}_y(\xi )$:

$$\begin{aligned} \Vert \nabla g_y(\xi )-\nabla g_y(\xi ')\Vert \le \Vert \xi -\xi '\Vert _*\,\,\forall \xi ,\xi ', \end{aligned}$$

(57)

where $\Vert \cdot \Vert _*$ is the norm conjugate to $\Vert \cdot \Vert$.

Indeed, since $\omega$ is strongly convex and continuously differentiable on $Y$, ${{\mathrm{Prox}}}_y(\cdot )$ is well defined, and from optimality conditions it holds

$$\begin{aligned} \langle \omega '({{\mathrm{Prox}}}_y(\xi ))+\xi -\omega '(y),{{\mathrm{Prox}}}_y(\xi )-z\rangle \le 0\,\,\forall z\in Y. \end{aligned}$$

(58)

Consequently, $g_y(\cdot )$ is well defined; this function clearly is convex, and the vector $-{{\mathrm{Prox}}}_y(\xi )$ clearly is a subgradient of $g_y$ at $\xi$. If now $\xi ',\xi ''\in F$, then, setting $z'={{\mathrm{Prox}}}_y(\xi ')$, $z''={{\mathrm{Prox}}}_y(\xi '')$ and invoking (58), we get

$$\begin{aligned} \langle \omega '(z')+\xi '-\omega '(y),z'-z''\rangle \le 0,\,\,\langle -\omega '(z'')-\xi ''+\omega '(y),z'-z''\rangle \le 0 \end{aligned}$$

whence, summing the inequalities up,

$$\begin{aligned} \langle \xi '-\xi '',z'-z''\rangle \le \langle \omega '(z')-\omega '(z''),z''-z'\rangle \le -\Vert z'-z''\Vert ^2, \end{aligned}$$

implying that $\Vert z'-z''\Vert \le \Vert \xi '-\xi ''\Vert _*$. Thus, a subgradient field $-{{\mathrm{Prox}}}_y(\cdot )$ of $g_y(\cdot )$ is Lipschitz continuous with constant 1 from $\Vert \cdot \Vert _*$ into $\Vert \cdot \Vert$, whence $g_y$ is continuously differentiable and (57) takes place. $\square$

2 $^0$. To derive Lemma 3 from Lemma 4, note that $f_y(x)$ is obtained from $g_y(\cdot )$ by affine substitution of variables and adding linear form:

$$\begin{aligned} f_y(x)=g_y(\gamma [{G(y)}- A^*x])+\gamma \langle a,x\rangle {+\omega (y)-\langle \omega '(y),y\rangle .} \end{aligned}$$

whence $\nabla f_y(x)=-\gamma A\nabla g_y(\gamma [{G(y)}-A^*x])+\gamma a=\gamma A {{\mathrm{Prox}}}_y(\gamma [{G(y)}-A^*x])+\gamma a$, as required in (39), and

$$\begin{aligned}&\Vert \nabla f_y(x')-\nabla f_y(x'')\Vert _{E,*}\\&\quad =\gamma \Vert A\left[ \nabla g_y(\gamma [{G(y)}-A^*x'])-\nabla g_y(\gamma [{G(y)}-A^*x''])\right] \Vert _{E,*}\\&\quad \le (\gamma L_A)\Vert \nabla g_y(\gamma [{G(y)}-A^*x'])-\nabla g_y(\gamma [{G(y)}-A^*x''])\Vert \\&\quad \le (\gamma L_A)\Vert \gamma [{G(y)}-A^*x']-\gamma [{G(y)}-A^*x'']\Vert _*\\&\quad \le (\gamma L_A)^2\Vert x'-x''\Vert _E=(L_A/L_G)^2\Vert x'-x''\Vert _E \end{aligned}$$

[we have used (57) and equivalences in (38)], as required in (40). $\square$

1.5 Review of Conditional Gradient algorithm

The required description of CGA and its complexity analysis are as follows.

As applied to minimizing a smooth – with Lipschitz continuous gradient

$$\begin{aligned} \Vert \nabla f(u)-\nabla f(u')\Vert _{E,*}\le \mathcal{L}\Vert u-u'\Vert _E,\,\,\forall u,u'\in X, \end{aligned}$$

convex function $f$ over a convex compact set $X\subset E$, the generic CGA is the recurrence of the form

$$\begin{aligned} \begin{array}{rcl} u_1&{}\in &{} X\\ u_{s+1}&{}\in &{} X \,\,\hbox { satisfies }\,\, f(u_{s+1})\le f(u_s+\gamma _s [u_s^+-u_s]),\,s=1,2,\ldots \\ \gamma _s&{}=&{}{2\over s+1},\,u_s^+\in \mathop {\hbox {Argmin }}\nolimits _{u\in X}\langle f'(u_s),u\rangle .\\ \end{array} \end{aligned}$$

The standard results on this recurrence (see, e.g., proof of Theorem 1 in [15]) state that if $f_*=\min _X f$, then

$$\begin{aligned} \begin{array}{ll} (a)&{}\epsilon _{t+1}:=f(u_{t+1})-f_*\le \epsilon _{t}-\gamma _t\delta _t+{2\mathcal{L}R^2\gamma _t^2},\,t=1,2,...\\ &{}\delta _t:=\max \nolimits _{u\in X} \langle \nabla f(u_t),u_t-u\rangle ;\\ (b)&{}\epsilon _t\le {2\mathcal{L}R^2\over t+1},t=2,3,\ldots \\ \end{array} \end{aligned}$$

(59)

where $R$ is the smallest of the radii of $\Vert \cdot \Vert _E$-balls containing $X$. From (59.$a$) it follows that

$$\begin{aligned} \gamma _\tau \delta _\tau \le \epsilon _\tau -\epsilon _{\tau +1}+2\mathcal{L}R^2\gamma _\tau ^2,\,\tau =1,2,\ldots ; \end{aligned}$$

summing up these inequalities over $\tau =t,t+1,\ldots ,2t$, where $t>1$, we get

$$\begin{aligned} \left[ \min _{\tau \le 2t}\delta _\tau \right] \sum _{\tau =t}^{2t} \gamma _\tau \le \epsilon _t+2\mathcal{L}R^2\sum _{\tau =t}^{2t}\gamma _\tau ^2, \end{aligned}$$

which combines with (59.$b$) to imply that

$$\begin{aligned} \min _{\tau \le 2t}\delta _\tau \le O(1)\mathcal{L}R^2{{1\over t}+\sum _{\tau =t}^{2t}{1\over \tau ^2}\over \sum _{\tau =t}^{2t}{1\over \tau }}\le O(1){\mathcal{L}R^2\over t}. \end{aligned}$$

It follows that given $\epsilon <\mathcal{L}R^2$, it takes at most $O(1){\mathcal{L}R^2\over \epsilon }$ steps of CGA to generate a point $u^\epsilon \in X$ with $\max _{u\in X} \langle \nabla f(u^\epsilon ),u^\epsilon -u\rangle \le \epsilon$.